462 

 Xi = X.2 sec^ (p, ?/i = ^2, ,„.v 



the equation of the cylinderj which is hyperboHc, being 



x* tan^ ^- z^= -K. (26) 



The focal and dirigent are each a pair of right lines pa- 

 rallel to the axis of the cylinder ; the corresponding lines 

 passing through a focus and the adjacent directrix of any 

 section perpendicular to the axis. The directive planes are 

 parallel to the asymptotic planes of the cylinder. 



In this case, if k = 0, the surface is reduced to two di- 

 rective planes, and the focal and dirigent to the intersection 

 of these planes. 



III. When mzz I, and ^ = 0, both a and b vanish, and 

 the surface is the parabohc cylinder. If, as is allowable, 

 we suppose g and k to vanish, the equation of the cylinder 

 becomes 



s2 + 2Hy=:0, (27) 



and we have 



n = y2-yu x,-x^, 

 X^^y^-x,^-y,^ = Q; ^ ^ 



whence 



y, = - 1h, ^2 = |H. (29) 



The focal and dirigent are each a right line parallel to the 

 axis of X, the former passing through the focus, the latter 

 meeting the directrix of the parabolic section made by the 

 plane oiyz. The plane oi xy is the directive plane. 



§ 7. We learn from this discussion, that, among the surfaces 

 of the second class, the hyperbolic paraboloid is the only 

 one which admits a twofold modular generation ; the modu- 

 lus, however, being the same for both its focals. In the 

 elliptic paraboloid the modular focal is restricted to the plane 

 of that principal section which has the greater parameter ; 

 we shall therefore suppose a parabola to be described in 

 the plane of the other principal section, according to the 



